There already is a great answer on the question, I just want to add some of my thoughts. (It will contain some duplication from the previous answer, so if this is a bad thing just let me know and I'll remove it :) ).
The Responsibility of ID generation can belong to different part of an organization or a system.
Sometimes the ID will be generated by some special rules like a Social Security Number. This number can be used for ID of a Person in a system, so before creating a Person entity this code will have to be generated from a specific SSNGenerator Service.
We can use a random generated ID like a UUID. UUIDs can be generated outside of the Repository and assigned to the entity during creation and the Repository will only store it (add, save) it to the DB.
IDs generated by databases are very interesting. You can have Sequential IDs like in RDBMS, UUID-ish like in MonogoDB or some Hash. In this case the Responsibility of ID generation is assigned to the DB so it can happen only after the Entity is stored not when it's created. (I'm allowing myself freedom here as you can generate it before saving a transaction or read the last one etc.. but I like to generalize here and avoid discussing cases with race conditions and collisions). This means that you Entity does't have an identity before the save completes. Is this a good thing? Of course It depends :)
This problem is a great example of leaky abstractions.
When you implement a solution sometimes the technology used will affect it. You will have to deal with the fact that for example the ID is generated by your Database which is part of your Infrastructure (if you have defined such a layer in your code). You can also avoid this by using s UUID even if you use a RDBMS, but then you have to join (again technology specific stuff :) ) on these IDs so sometimes people like to use the default.
Instead of having Add or AddAndCreate you can have Save method instead that does the same thing, it's just a different term that some people prefer. The repository is indeed often defined as an "In memory collection" but that doesn't mean that we have to stick to it strictly (It can be a good thing to do that most of the time but still...).
As mentioned, if you database generates ID's, the Repository seems like a good candidate to assign IDs (before of after storing) because it is the one talking to the DB.
If you are using events the way you generate ID's can affect things. For example lets say you want to have UserRegisteredEvent with the UserID as s property. If you are using the DB to generate ID's you will have to store the User first and then create and store/dispatch the event or do something of the sort. On the other hand if you generate the ID beforehand you can save the event and the entity together (in a transaction or in the same document doesn't matter). Sometimes this can get tricky.
Background, experience with technologies and framework, exposure to terminology in literature, school and work affects how we think about things and what terminology sounds better to us. Also we (most of the time) work in teams and this can affect how we name things and how implement them.
int
asId
? Who's able to generate a valid one before inserting the entity? – Chancelor